Symmetry defect of $n$-dimensional complete intersections in $\mathbb{C}^{2n-1}$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929330375884800 |
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| author | Dias, L. R. G. Jelonek, Z. |
| author_facet | Dias, L. R. G. Jelonek, Z. |
| contents | Let $X, Y \subset \mathbb{C}^{2n-1}$ be $n$-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point $x \in X$ to a point $y \in Y$. This set is defined as the image of the map $Φ(x,y)=\frac{x+y}{2}.$ Under geometric conditions on $X$ and $Y$, we prove that the symmetry defect of $X$ and $Y$, which is the bifurcation set $B(X,Y)$ of the mapping $Φ$, is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set $B(X,Y)$ and we present an estimate for its degree. Moreover, for any two $n$-dimensional strong complete intersections $X,Y\subset \mathbb{C}^{2n-1}$ (including the case $X=Y$) we introduce a generic symmetry defect set $\tilde{B}(X,Y)$ of $X$ and $Y$, which is defined up to homeomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_18927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetry defect of $n$-dimensional complete intersections in $\mathbb{C}^{2n-1}$ Dias, L. R. G. Jelonek, Z. Algebraic Geometry Differential Geometry Let $X, Y \subset \mathbb{C}^{2n-1}$ be $n$-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point $x \in X$ to a point $y \in Y$. This set is defined as the image of the map $Φ(x,y)=\frac{x+y}{2}.$ Under geometric conditions on $X$ and $Y$, we prove that the symmetry defect of $X$ and $Y$, which is the bifurcation set $B(X,Y)$ of the mapping $Φ$, is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set $B(X,Y)$ and we present an estimate for its degree. Moreover, for any two $n$-dimensional strong complete intersections $X,Y\subset \mathbb{C}^{2n-1}$ (including the case $X=Y$) we introduce a generic symmetry defect set $\tilde{B}(X,Y)$ of $X$ and $Y$, which is defined up to homeomorphism. |
| title | Symmetry defect of $n$-dimensional complete intersections in $\mathbb{C}^{2n-1}$ |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2404.18927 |